Let $V$ be an irreducible affine variety in $\mathbb{C}^n$. Is it true that $V$ is locally path-connected?
I am trying to build on https://math.stackexchange.com/a/2628615/416079 that shows that $V$ is path-connected. Copying the relevant part here:
Let $p:\tilde V\to V$ be a desingularization of $V$.
For $a,b\in V$ there exist $\tilde a, \tilde b\in \tilde V$ with $p(\tilde a)=a, p(\tilde a)=b$.
Join $\tilde a, \tilde b\in \tilde V$ by a path $\gamma:[0,1]\to \tilde V$ (possible since $\tilde V$ is a connected complex manifold).
The path $p\circ \gamma$ then joins $a$ to $b$, qed.
Can I adapt the proof as follows?
Take any $a \in V$ and $W\subset V$ be an open subset of $V$ that contains $a$.
a) Then $p^{-1}(W)$ is an open set in $\tilde V$ that contains $\tilde a = p^{-1}(a)$. - Is this true?
b) The associated analytic variety of $\tilde V$ (now referring to https://math.stackexchange.com/a/1714997/416079) is a connected complex manifold, so it is locally path-connected. Hence, there exists a path-connected open set $\tilde a \in \tilde U \subset p^{-1}(W)$. Then $p(\tilde U)$ is an open set in $V$ and $p(\tilde U)\subset W$. - Is this true?
c) Furthermore, any path $\tilde \gamma \subset \tilde U$ maps to a path $\gamma \subset p(\tilde U)$. - Is this true?
d) Therefore, $p(\tilde U)$ is a path-connected neighbourhood of $a$ that is contained in $W$, so $V$ is locally path-connected.
Is it true in general that this map $p$ preserves the connectedness properties?
My end goal is to prove that if $V$ is an irreducible affine variety and $Y\subsetneq V$ is a variety with $\dim Y < \dim V$, then $V \backslash Y$ is path-connected. So any hints as to whether this is actually true or possible alternative ways to prove it would be highly appreciated.